Let $f$ be a real-valued function at $x\in\mathbb{R}$, we state that $f$ is Lipschitz continuous at $x$ if there exists a constant $C$ such that : $$ \|f(y)-f(x)\|\leqslant C\|y-x\| $$ for all $y\in\mathbb{R}$ which is sufficiently near $x$.
We can rewrite the equation above to be : $$ \|\nabla f\|\triangleq\frac{\|f(y)-f(x)\|}{\|y-x\| }\leqslant C $$ My question is: For a several variable Lipschitz function $f$ at $\mathbb{x}\in\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ if there exists a constant $L$ such that : $$ \frac{\|f(\mathbf{y})-f(\mathbf{x})\|}{\|\mathbf{y}-\mathbf{x}\| }\leqslant L $$ can we simply acknowledge that : $$ \frac{\|f(\mathbf{y})-f(\mathbf{x})\|}{\|\mathbf{y}-\mathbf{x}\| }\qquad\text{is}\qquad \|\nabla f(\mathbf{x},\mathbf{y})\|\;?\tag{1} $$ where $\|.\|$ can be any norm (I will consider $L_{2}$ norm).